Download Chapter 6 One-Electron Reduction Potentials of Aqueous Co2+

Chapter 6
One-Electron Reduction Potentials of Aqueous Co2+, Ni2+, Cu2+, and
Zn2+ Measured with Gas-Phase Electrochemistry
Demireva, M; Williams, E. R. “One-Electron Reduction Potentials of Aqueous Co2+, Ni2+, Cu2+, and Zn2+ Measured
with Gas-Phase Electrochemistry”
6.1 Abstract
Absolute one-electron reduction potentials for aqueous Co2+, Ni2+, Cu2+, and Zn2+ were
determined from gas-phase electrochemistry experiments performed on these ions contained in
aqueous nanodrops. The one-electron reduction potentials have not previously been measured
with conventional solution-phase electrochemistry methods for these ions except for Cu2+.
Absolute reduction enthalpies for these metal ions with between 36 and up to 240 water
molecules attached are measured and are extrapolated to infinite cluster size. From these
measurements, absolute one-electron reduction potentials of 2.24, 3.00, 4.04, and 1.49 V are
obtained for Co2+, Ni2+, Cu2+, and Zn2+, respectively. An absolute value for the SHE potential of
3.88 V is obtained from the measured absolute reduction potential of Cu2+. This value is slightly
lower than that reported previously from measurements of Eu3+ (4.11 V) using the same method,
likely owing to errors in modeled water binding energies, which can ultimately be eliminated
with results from laser calibration experiments. Relative one-electron reduction potentials for
Co2+, Ni2+, and Zn2+ referenced relative to the SHE potential of 0.00 V are -1.64, -0.88 and -2.39
V, respectively. These values are likely more accurate than the absolute values because errors
from water binding energies should largely cancel. These results demonstrate the advantages of
using gas-phase electrochemistry to measure reduction potentials of ions that are difficult to
measure with traditional solution-phase electrochemistry methods.
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6.2 Introduction
In solution-phase electrochemistry experiments, reduction or oxidation (red-ox) is
controlled by a potential and red-ox potentials are measured relative to other half-cell potentials.
For multiple electron transfer processes, individual red-ox potentials for each electron transfer
can be measured if addition or removal of the sequential electron requires more energy (i.e.,
normal ordering of potentials).1 In cases where it is energetically much more favorable to add or
remove a subsequent electron (i.e., potential inversion), rapid sequential or concerted electron
transfers occur such that the concentration of the intermediate resulting from the first electron
transfer is negligible.1 In these cases, measuring a red-ox potential for the intermediate electron
transfer process using conventional solution-phase electrochemistry methods is difficult.
Because red-ox potentials in solution are measured relative to other half-cell potentials, a
reference potential is required. The universally accepted reference potential is the standard
hydrogen electrode (SHE) potential, which has arbitrarily been assigned 0 V. Many different
methods have been developed to obtain an absolute SHE value and thereby establish an absolute
electrochemical scale.2-9 However, these approaches have been indirect and a relatively wide
range of values for the absolute SHE potential has been reported. 2-9 A definition for the absolute
1
SHE potential is H+ (aq) + e− → H(g) referenced to an electron at infinite distance.9 If the
2
SHE value obtained from this definition includes the contribution from the surface potential of
water in the solvation energy of the proton it is known as a “real” potential; if not, the value
corresponds to the “absolute” SHE potential. The surface potential of water is likely a small
value,10,11 and thus the difference between the “real” and “absolute” SHE values should be small.
In contrast to solution-phase electrochemistry measurements, the absolute gas-phase
ionization energies of atoms and ions can easily be measured. These ionization potentials can be
related to solution-phase red-ox potentials through thermodynamic relations that include the
solvation energies of the reduced and oxidized species. Many methods have employed this
approach to obtain absolute and relative potentials. 12-20 The difficulty with these methods is to
accurately account for the ion solvation energies of the reduced and oxidized species. This is
because the interactions between the ions and the solvent molecules in the first solvation shells
can be complex and these solvent molecules can behave anomalously compared to bulk. 15-17,21,22
For example, calculations have shown that ligand field splitting stabilization energies can
contribute significantly to the hydration energies of first row transition metal ions and when
these effects are included calculated values are better able to reproduce experimental data. 21
Generally, more accurate solvation energies can be obtained by explicitly taking into account
quantum-mechanical interactions between the ions and the solvent molecules in the first few
solvation shells.16-18,21,23
Alternatively, gas-phase studies of clusters can be used to gain insight into the physical
properties of ions in the condensed-phase.24-29 In such studies, where the number of solvent
molecules can be precisely controlled, information about the effects of solvent on the stability
and reactivity of ions can be obtained and related to properties in bulk solution. 29-38 For example,
dissociation experiments of hydrated metal ion clusters indicate that the critical cluster size at
which the charge separation reaction occurs to form the metal hydroxide and protonated water is
correlated with the corresponding hydrolysis constants of the metal ions in bulk solution. 35-37
Other desolvation experiments of hydrated metal ions in “unusual” charge states (e.g., Mg +, Al+,
or V+ ) show that charge transfer reactions that occur at ion-specific cluster sizes oxidize the
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metal ions to their preferred solution-phase valence states (i.e., Mg2+, Al3+ , and V2+ and V3+).29-31
Related experiments indicate that Mg+, Al+, and V+ are also oxidized in the presence of HCl in
the same aqueous nanodrop, whereas Cr+, Mn+, Fe+, Co+, Ni+, and Cu+ do not change their
oxidation states and instead form a metal chloride (MCl) “precipitate,” consistent with the
propensity of these metal ions to be oxidized in solution.32 Cluster measurements can also be
extrapolated to infinite size for comparison with bulk values or to infer properties of ions in bulk
solution. For example, vertical detachment energies extrapolated to infinite size for water
clusters containing an excess electron have been used to assign, although with some debate,
structural isomers for the hydrated electron (i.e., surface or internally solvated states) in these
experiments.26,39-41
Our group has relatively recently introduced a gas-phase electrochemistry method that
measures absolute reduction potentials of ions contained in aqueous nanodrops. 42-47 In these
experiments, capture of a thermally generated electron by a hydrated metal ion cluster results in
evaporation of water molecules. The hydrated cluster serves as a “nanocalorimeter”, where the
energy deposited, i.e. recombination energy, is determined by accounting for the binding
energies and the energy partitioned into translations, rotations and vibrations of each lost water
molecule.47-49 The absolute recombination energies obtained from these cluster measurements
can be related to the absolute reduction potentials of the metal ions in bulk aqueous solution.
This gas-phase electrochemistry technique has been used to obtain an absolute value for the SHE
potential of 4.05 V,46 4.11 V,45 and 4.21 V47 from three largely independent approaches.44 The
most recent approach extrapolates cluster recombination enthalpies of hydrated Eu 3+ to infinite
size to obtain the absolute one-electron reduction potential in bulk aqueous solution.45 An
absolute value for the SHE potential is then obtained from the absolute reduction potential of
Eu3+ and its measured relative value from solution-phase experiments. Unlike the other two
approaches, this method does not require a solvation model, and is thus the most direct and
potentially also the most accurate.
Reduction in these gas-phase electrochemistry experiments depends on the low
probability of capturing a gaseous electron by a hydrated ion cluster and is not controlled by a
potential as in solution-phase experiments. Thus, our gas-phase electrochemistry method can be
used to measure one-electron reduction potentials in cases where potential inversion occurs. 1
Such behavior is known for many of the first row divalent transition metal ions, including Co 2+,
Ni2+ and Zn2+,1 which therefore do not have measured one-electron reduction potentials. Some
divalent metal ions, such as Mg2+ and Ca2+, do not have stable corresponding monovalent ions in
water, as also indicated by gas-phase cluster studies where Mg+(H2O)n and Ca+(H2O)n consist of
a separately solvated doubly charged metal ion and an electron within the same nanodrop, i.e., an
ion-electron pair.30,50-52 However, Co+, Ni+ and Zn+ have previously been produced and observed
in aqueous solution with pulse radiolysis and spectrophotometric techniques, indicating that
Co2+, Ni2+, and Zn2+ can be directly reduced by a hydrated electron. 53,54 This is also supported by
gas-phase electron capture experiments of Co2+, Ni2+, and Zn2+ with 32 water molecules, which
show that direct metal ion reduction occurs upon electron capture in these nanodrops. 48
Here, absolute one-electron reduction enthalpies of size selected M2+(H2O)n for M = Co,
Ni, Cu, and Zn with n = 36 – 240 are measured using gas-phase electrochemistry and ion
nanocalorimetry. These cluster measurements are extrapolated to infinite size to obtain the
absolute one-electron reduction enthalpies of the corresponding metal ions at infinite dilution in
aqueous solution. The measured absolute reduction enthalpies are combined with an
experimental value for the absolute reduction entropy of Cu 2+ to obtain absolute one-electron
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reduction potentials. An absolute value for the SHE potential is also obtained from the absolute
reduction potential for Cu2+ reported here and its relative potential from solution-phase
experiments. Relative one-electron reduction potentials for Co2+, Ni2+ and Zn2+ referenced to the
SHE potential are determined and are to our knowledge the first experimentally reported values.
6.3 Methods
6.3.1 Experiments. Electron capture (EC) and ion nanocalorimetry experiments were
performed using a 7 T Fourier transform ion cyclotron resonance mass spectrometer equipped
with a nanoelectrospray ionization (nESI) source and a temperature controlled ion cell. The
experimental apparatus is described in detail elsewhere, and has been modified by incorporating
a 7.0 T magnet and improved vacuum chamber. 45,55-57 Briefly, hydrated metal ion clusters are
generated from 5 – 10 mM aqueous solutions containing the metal salts (CoSO 4, NiCl2, CuSO4,
or ZnCl2) using nESI and borosilicate capillaries pulled to an inner tip diameter of ~2 μm. A
potential of ~400 – 800 V relative to the metal entrance capillary of the mass spectrometer is
applied to a platinum wire that is in contact with the solution. Ions are introduced into the mass
spectrometer through a heated (~40 – 60 °C) entrance capillary and guided through five stages of
differential pumping into the ion cell. The source, trapping, and excite and detect parameters are
adjusted to optimize the signal for the clusters of interest. The ion cell is surrounded by a copper
jacket with thermocouples and is temperature regulated (133 K) with a controlled flow of liquid
nitrogen.56 Ions are accumulated in the cell for 5 – 13 s, during which time dry nitrogen is
introduced to enhance trapping and thermalization of the ions. The ions are subsequently stored
for 8 – 20 s to ensure that the pressure returns below 10-8 Torr and that the ions have steady-state
internal energies. Individual hydrated ion precursors are isolated for M 2+(H2O)n with n < 100 and
distributions consisting of five adjacent precursor ions are isolated for n ≥ 100 using stored
waveform inverse Fourier transform excitation to eject all other ions from the cell. The ensemble
measurements used for the larger clusters improve the precision of the nanocalorimetry
experiments.58 Following isolation of the precursor(s), there is a 40 ms delay before electrons are
introduced into the ion cell using a heated dispenser cathode (~3.4 A, ~5.8 V) (HeatWave Labs,
Inc., Watsonville, CA) mounted axially with respect to the cell center. A Cu mesh mounted ~0.5
cm in front of the cathode is held at +9.2 V. Electrons are introduced into the cell for 120 ms by
applying a negative housing potential to the cathode (-3.2 to -4.2 V and adjusted to maximize
product ion abundance). A potential of +10 V is applied to the housing of the cathode at all other
times to prevent electrons from entering the cell. A delay of 0.5 or 1.0 s for n less than or greater
than ~60, respectively, is used after electron irradiation, but before excitation and detection to
allow time for reduced precursor ions to dissociate. 45,57 The ions are subsequently excited and
detected using a MIDAS data system.59 The average number of water molecules lost from a
single isolated precursor ion is obtained from a weighted average of the product ion abundances
and is corrected for dissociation from blackbody infrared radiative dissociation (BIRD) to give
the average number of water molecules lost due to EC alone. For a distribution of isolated
adjacent precursor ions, the average water molecule loss upon EC is obtained from the difference
in the average cluster sizes between the precursor and product ion distributions. 60
6.3.2 Calculations. Ion internal energies for calculations of the recombination energies
are obtained as described previously using calculated harmonic frequencies for an energy
minimized B3LYP/LACVP**++ structure of Ca 2+(H2O)14.45 Internal energy for a specific cluster
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size is calculated by linearly scaling the degrees-of-freedom for Ca2+(H2O)14 by those of the
cluster of interest. The average water molecule loss measured in the EC experiments is converted
to a recombination energy (RE) by accounting for the binding energy and energy taken away by
each water molecule that is lost in the form of translations, rotations, and vibrations (E TRV).45-47,57
The binding energies are obtained from a discrete implementation of the Thomson liquid drop
model61 and the average ETRV is given by 5/2 kT* obtained from Klots’ water evaporation
model,62-64 where k is the Boltzmann constant and T* is the effective temperature of the cluster
ion prior to dissociation. The effective temperatures of the cluster ions formed upon EC and after
subsequent water molecule loss are modeled statistically by iteratively solving for the cluster
internal energies that give the observed average water molecule loss. E TRV distributions are
modeled with an exponential function which approaches zero at high energies:48
PTRV(E) = <E>-1exp(–E/<E>)
(6.1)
where <E> is the average ETRV value given by 5/2 kT*. For x number of water molecules lost, x
exponential functions are convolved to obtain the cumulative ETRV distribution for all of the lost
water molecules.
6.4 Results and Discussion
6.4.1 Electron Capture by M2+(H2O)n with M = Co, Ni, Cu, and Zn. Nanoelectrospray
ionization of aqueous solutions containing the Co, Ni, Cu, and Zn metal salts results in broad
distributions of hydrated metal ion clusters. The distribution of clusters can be shifted in size by
adjusting instrument parameters. Spectra illustrating generation of hydrated Co2+ with up to 400
water molecules under these separate set of conditions are shown in Figure 6.1. Larger clusters
are formed at lower source capillary temperatures and the excite parameters are optimized for ion
detection at lower frequencies (higher m/z).
A precursor ion cluster of interest is isolated and reacted with thermally generated
electrons for 120 ms. The product ions formed from EC by M2+(H2O)36 with M = Co, Ni, Cu,
and Zn are shown in Figure 6.2 (a) – (d), respectively. The horizontal axis has been converted to
a cluster size scale in these mass spectra to more clearly indicate the differences in water loss
between the metal ions for a given precursor ion cluster size. EC results in formation of reduced
precursor ions that lose on average 12.2, 13.5, 16.2, and 11.3 water molecules for Co 2+, Ni2+,
Cu2+, and Zn2+, respectively. Water molecule loss (25 ± 2%) also occurs from the precursor ions
(data not shown) due to absorption of blackbody photons, i.e., BIRD. The average number of
water molecules lost due to EC alone is obtained from a weighted average of the product ion
intensities, which is corrected for dissociation from BIRD that is estimated from the dissociation
of the precursor ions. For example, on average 12.2 water molecules are lost from the reduced
precursor of Co2+(H2O)36 and 0.2 water molecules are lost from this precursor ion due to BIRD,
resulting in an average number of 12.2 – 0.2 = 12.0 water molecules lost due to EC alone. For
Ni2+(H2O)36, Cu2+(H2O)36, and Zn2+(H2O)36, EC results in loss of on average 13.2, 16.0, and 11.0
water molecules, respectively.
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Figure 6.1. Representative nESI mass spectra for Co 2+(H2O)n with instrument parameters tuned for (a) small, (b)
mid-size, and (c) large hydrated clusters.
For larger clusters with n ≥ 100, a distribution of five adjacent precursor ions was isolated
for improved precision in the water loss measurements. 58 Representative EC product ion mass
spectra for isolated precursor ion distributions with n = 108 – 112 and n = 208 – 212 for Co2+,
Ni2+, Cu2+, and Zn2+ are shown in Figure 6.2 (e) – (h) and Figure 6.2 (i) – (l), respectively. EC of
isolated precursor ion distributions with n = 108 – 112 results in formation of product ion
distributions with n = 96 – 101 for Co2+ (Figure 6.2 (e)), n = 94 – 100 for Ni2+ (Figure 6.2 (f)), n
= 92 – 97 for Zn2+ (Figure 6.2 (g)), and n = 96 – 103 for Cu2+ (Figure 6.2 (h)). The average
number of water molecules lost due to EC alone is obtained directly from the difference in the
average cluster sizes of the precursor and product ion distributions from these mass spectra. 60 For
example, the average precursor cluster size for Co 2+(H2O)n with n = 108 – 112 isolated is 108.6
(n = 103 – 112, lower cluster sizes formed by BIRD) and the average product cluster size is 98.7
(n = 96 – 101), which gives an average water molecule loss of 108.6 – 98.7 = 9.9. The
corresponding average water molecule losses for Ni 2+, Cu2+, and Zn2 are 11.5, 13.9, and 8.5,
respectively. EC of M2+(H2O)208-212 results in an average water molecule loss of 8.4, 10.1, 13.0,
and 7.3 for M = Co, Ni, Cu, and Zn, respectively (Figure 6.2 (i) – (l)). EC by these four divalent
transition metal ions results in a different number of water molecules lost for a given precursor
ion cluster size, indicating that these metal ions capture an electron and are reduced to the 1+
charge state in the nanodrops. Loss of the same number of water molecules for each metal ion
would indicate that an ion-electron pair is formed, where the charge state of the metal ion
remains 2+ and the captured electron is solvated by water molecules in the nanodrop. 50,57
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Figure 6.2. Electron capture mass spectra (product ion region) of M2+(H2O)36 (left) with M = (a) Co, (b) Ni, (c) Cu,
and (d) Zn, M2+(H2O)108-112 (middle) with M = (e) Co, (f) Ni, (g) Cu, and (h) Zn, and M 2+(H2O)208-212 (right) with M
= (i) Co, (j) Ni, (k) Cu, and (l) Zn. Electronic noise peaks are marked by asterisks (*) and the horizontal axis is
converted to a cluster size scale to more readily compare effects of ion identity.
6.4.2 Factors that Affect the Extent of Water Molecule Loss. The average number of
water molecules lost from EC of size-selected Co2+(H2O)n, Ni2+(H2O)n, Cu2+(H2O)n, and
Zn2+(H2O)n was obtained between n = 36 and 240 and these data are shown in Figure 6.3. The
average number of water molecules lost depends on both the precursor ion cluster size as well as
the metal ion identity. The average number of water molecules lost decreases with increasing
precursor ion cluster size for each metal ion. This effect is predominantly due to increasing
solvation of the metal ion with increasing hydration extent, which results in a lower
recombination energy upon EC.45,50,57 For a given precursor ion cluster size, the ordering of most
to least water molecules lost upon EC for these four divalent transition metal ions is Cu 2+ > Ni2+
> Co2+ > Zn2+. This trend is consistent with previous EC results for these metal ions with 32
water molecules attached.48
Previous results on EC by Ca2+(H2O)15 and Ca2+(H2O)32 show that the initial kinetic
energy of the electrons does not significantly affect the extent of water molecule loss observed
under these experimental conditions.55 The effects of the initial electron kinetic energy on larger
clusters in which the metal ion is reduced was investigated by varying the cathode housing
potential between -3.2 V and -5.7 V in 0.5 V increments. For Co2+(H2O)52, an average of 11.03
and 11.06 water molecules are lost from the reduced precursor ion at a cathode housing potential
of -3.2 V and -5.7 V, respectively (Figure 6.4 (a)). Extrapolating the average water molecule loss
data in Figure 6.4 (a) to a housing potential of 0 V results in a value of 11.1 ± 0.1. This
uncertainty is a measure of the precision in the measured values and is in good agreement with
the previous results.55 The negligible change in water molecule loss over this limited cathode
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potential range is attributed to EC having highest cross section when the relative kinetic energy
of the ion and electron approaches zero.55 The electrons are trapped and both the ion and electron
have a range of kinetic energies owing to energy loss and the influence of electric fields inside
the ion cell.
Figure 6.3. The average number of water molecules lost as a result of EC by M2+(H2O)n with M = Cu (squares), Ni
(diamonds), Co (circles), and Zn (triangles) as a function of the precursor ion cluster size, n. Data for n = 32 have
previously been published.48
For small clusters, dissociation upon EC is fast compared to the time scale of the
experiment.45,57 However, the number of internal degrees-of-freedom increases with increasing
cluster size and the recombination energy decreases due to improved ion solvation. Both of these
factors contribute to a decrease in the effective temperature of the reduced precursor ion and a
slower dissociation rate.45,57 To investigate the time required for complete dissociation of the
larger clusters, EC experiments on Co2+(H2O)~200 were performed, where the reaction delay
following EC was varied. The average number of water molecules lost as a function of
dissociation time for Co2+(H2O)~200 are shown in Figure 6.4 (b). An average of 8.6 and 8.8 water
molecules are lost after a reaction delay of 40 ms and 500 ms, respectively. There is no
significant change in the number of water molecules lost at longer delay times, indicating that
dissociation as a result of EC is complete before ~0.5 s. Thus, a time delay after electron
irradiation of 0.5 s or 1 s was used for n less than or greater than ~60, respectively, to ensure that
any kinetic shift effects are negligible.
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Figure 6.4 (a) Average number of water molecules lost upon reduction of Co 2+(H2O)52 as a function of the heated
cathode housing potential. The red dashed line corresponds to the extrapolated least squares best fit to the data with
a standard uncertainty in the vertical intercept of ± 0.1. (b) The average number of water molecules lost from
Co2+(H2O)~200 upon EC as a function of the reaction delay after electron irradiation but before ion excitation and
detection, showing kinetic shift effects.
6.4.3 Recombination Energy (RE). REs are obtained from the average number of water
molecules lost upon EC by a precursor ion or ions by accounting for the individual water
molecule binding energies and the average E TRV of each evaporated water molecule.47 For larger
clusters, these REs are adiabatic values because the water molecule reorganization as a result of
the change in ion charge state occurs on a faster time scale (ps) than loss of the first water
molecule, although this is not always the case for small clusters that have high RE values.48
Capture of an electron likely forms a high energy Rydberg state that can rapidly relax to form a
vibrationally “hot” ion. The effect on ion internal energy is illustrated in Figure 6.5 (top), where
the initial internal energy of the precursor ion increases by the RE upon EC. The activated
precursor ion dissipates this excess internal energy rapidly by sequential evaporation of water
molecules illustrated in Figure 6.5 (bottom). The internal energy distribution shifts to lower
energy with each water molecule that is lost by an amount corresponding to the sum of the
binding energy and the average E TRV, until the final product ion is cooled to an effective
temperature corresponding to the initial temperature of the experiment. The amount of energy
that is partitioned into ETRV depends on the effective temperature of the reduced precursor. 47,48
The effective temperature of the reduced precursor and that of the resulting product ions after
each subsequent water molecule that is lost are modeled statistically by iteratively solving for the
cluster internal energies (illustrated in Figure 6.5 bottom) that give the observed average water
molecule loss.45,48
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Figure 6.5. Schematic of the modeled internal energy evolution of an activated reduced precursor ion cluster. Full
internal conversion of the energy deposited upon EC results in an increase in the effective temperature of the
reduced precursor (top) and sequential water molecule loss subsequently dissipates the excess energy (bottom). The
recombination energy is obtained by accounting for the binding energies and the energy lost to translational,
rotational, and vibrational modes (E TRV) of the lost water molecules. The broadening of the internal energy
distribution with each water molecule lost is due to ETRV, which is modeled with a simple exponential distribution.
Even though these hydrated metal ion clusters lose a significant number of water
molecules upon EC, e.g., up to 17 water molecules from Cu 2+(H2O)36 (Figure 6.2 (c)), the
product ion distributions are very narrow, consisting only of at most three product ions from an
individual precursor ion. Previous results for large clusters indicate that the width of the product
ion distributions in the EC experiments where BIRD is low can be entirely accounted for by the
ETRV of the water molecules that are lost,48 consistent with a singular value of energy deposited
(the RE) upon metal ion reduction.
6.4.4 Effects of ETRV. The effect of ETRV on the width of the product ion distributions as
a function of precursor ion cluster size for Cu2+(H2O)n with n = 36, 42, 56, and 90 is shown in
Figure 6.6 (a) – (d), respectively. The relative product ion abundances (black sticks) from the
experimental mass spectra are plotted as a function of the cumulative binding energies needed to
form these product ions (bottom horizontal axis), which are corrected for water molecule loss
due to BIRD. The corresponding number of water molecules lost is indicated on the top
horizontal axis. For example, the black sticks in Figure 6.6 (a) correspond to the experimental
product ion abundances obtained from the mass spectrum in Figure 6.2 (c) for Cu2+(H2O)36. The
cumulative binding energies required to form these product ions are less than the REs (vertical
dashed line) because of the energy carried away as E TRV by the water molecules that are lost. The
ETRV distributions are plotted as RE – PTRV,x to illustrate that fewer water molecules are lost than
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would be expected if no energy partitioned into E TRV but was used to only break the water-water
bonds. The ETRV distribution for the first water molecule lost, PTRV,1, is exponential with the most
probable ETRV value of 0 (this approximation neglects effects of conservation of angular
momentum). The cumulative ETRV distribution for the second water molecule is a convolution of
two exponentials, and indicates that the probability of losing both the first and second water
molecule with no ETRV is nearly zero. Increasing the number of convolved E TRV distributions
results in a shift to larger average energy values and broadens the E TRV distributions. Included for
comparison are Gaussian distributions (dashed curves) obtained by fitting the experimental
product ion abundances.
6.4.5 Evidence for Non-Statistical Dissociation at Small Cluster Size. The agreement
between the experimental product ion distributions and the modeled cumulative ETRV
distributions varies with cluster size for Cu2+(H2O)n with n = 36, 42, 56, and 90 (Figure 6.6). The
Gaussian (experimental) fit for n = 36 (Figure 6.6 (a)) is significantly narrower with a full width
at half maximum (FWHM) of 0.5 eV than the modeled ETRV distribution with a FWHM of 0.9
eV. This indicates that the effective temperature for Cu 2+(H2O)36 upon EC is overestimated in the
statistical modeling, which results in a wider calculated product ion distribution than is observed
experimentally. Thus, non-statistical dissociation is likely occurring, where some water
molecules are lost prior to or on the timescale of internal conversion of the RE into vibrational
energy. This results in a lower initial effective temperature and thus a lower ETRV value. The
results for Cu2+(H2O)36 are consistent with previous EC results for Cu 2+(H2O)n with n = 24 and
32, where the modeled ETRV distributions are also narrower than the Gaussian fits to the
experimental product ion distributions by 0.5 and 0.4 eV in FWHM, respectively.48
In contrast, relatively good fits are obtained between the experimental and modeled E TRV
distributions for Co2+, Ni2+, and Zn2+ with n = 36 (Figure S6.1 in Supporting information) with
less than 0.1 eV difference in FWHM, consistent with earlier EC results for these metal ions with
n = 32.48 This is attributed in part to the lower RE values for these metal ions compared with that
for Cu2+, which result in slower water evaporation rates. Differences in inherent excited state
lifetimes for the different metal ions may also play a role. The non-statistical behavior observed
for Cu2+ with n = 32 and n = 36 could be the result of a multistep electronic-to-vibrational energy
conversion or the formation of a relatively long-lived excited state.48 With increasing cluster
size, relatively good fits are obtained between experimental and modeled E TRV distributions for
Cu2+(H2O)n with n = 42 (Figure 6.6 (b)) and n = 56 (Figure 6.6 (c)), with less than 0.1 eV
difference in FWHM between the Gaussian and ETRV fits. This indicates that internal conversion
for these larger clusters occurs faster than the loss of the first water molecule and that the width
of the observed product ion distributions can be fully accounted for by the modeled E TRV. This is
consistent with previous ETRV results,48 which show that the rate of water evaporation becomes
slower than internal conversion with increasing cluster size and decreasing RE. For Cu 2+(H2O)90
(Figure 6.6 (d)), the Gaussian fit to the experimental product ion distribution is significantly
broader than the modeled ETRV distribution. This is due to broadening in the product ion
distribution that arises from BIRD. The BIRD rate increases with increasing cluster size, 65 and
thus has a greater effect on the width of the experimental product ion distribution with increasing
size. This is also consistent with the larger BIRD correction with increased cluster size, as is
reflected in the larger difference between the average water molecule loss due to EC alone, i.e.
corrected for BIRD, and the average water molecule loss that would be obtained from the
weighted average of the product ion intensities with increasing cluster size (Figure 6.6).
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Figure 6.6. Experimental product ion abundances (black sticks) from EC by Cu 2+(H2O)n with (a) n = 36, (b) n = 42,
(c) n = 56, and (d) n = 90 as a function of the sum of the threshold dissociation energies required to form
Cu+(H2O)n-x (bottom horizontal axis), where x is the number of lost water molecules (top horizontal axis).
Cumulative ETRV distributions (solid lines) are obtained from convolving P TRV,x(E) = (5/2 kT*)-1exp(-E/(5/2 kT*) for
each lost water molecule. Product ion abundances are also fitted with a Gaussian function (dotted line) for
comparison. The calculated recombination energy from the average number of water molecules lost is indicated by
the black dashed line.
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6.4.6 Extrapolating Cluster Measurements to Bulk Solution. The gas-phase reduction
enthalpy of a hydrated metal ion cluster is related to the solution-phase reduction enthalpy of the
metal ion by the difference in the solvation enthalpies, ΔHSolv(Mz+) and ΔHSolv(M(z-1)+), of the
precursor and reduced precursor ions, respectively.45,47 This yields the following expression:
ΔHR(n) = ΔHAbs + ΔΔHSolv(n)
(6.2)
where ΔHR is the enthalpy of recombination for the hydrated metal ion cluster of size n, ΔHAbs is
the absolute reduction enthalpy for the metal ion in the condensed phase, and ΔΔHSolv(n) is the
solvation enthalpy difference between the precursor and reduced precursor ions for cluster size n.
The recombination enthalpy at 298 K is obtained from:45
ΔHR = –RE – cel
(6.3)
where cel is the integrated heat capacity for the electron from Fermi-Dirac statistics.66 To obtain
the absolute reduction enthalpy without relying on solvation models, these cluster measurements
can be extrapolated to infinite cluster size to obtain the energy of the corresponding process in
the condensed phase.45 The solvation enthalpy difference (ΔΔHSolv(n)) is proportional to the
inverse of the cluster radius, r, for a sufficiently large cluster that can be approximated as a
sphere. The volume of a cluster is proportional to the number of water molecules, n, which
results in r being proportional to n1/3, with the solvation enthalpy difference proportional to n-1/3.
By combining these proportionalities with equation 2, the recombination enthalpy is given by:
ΔHR(n) = ΔHAbs + C n-1/3
(6.4)
where C is a proportionality constant. Thus, for sufficiently large clusters that can be
approximated as spheres, ΔHR(n) should vary linearly with n-1/3 and have a vertical intercept
corresponding to ΔHAbs.45,57 This extrapolation method has been used previously to measure the
absolute one-electron reduction enthalpy of Eu3+,45 as well as the electron solvation enthalpy
using hydrated La3+ clusters, which form an ion-electron pair upon EC.57
The negative of ΔHR(n) as a function of n-1/3 for M2+(H2O)n with M = Co, Ni, Cu, and Zn
is shown in Figure 6.7. The data are fit with lines for n = 32 – 240 for all metal ions except for
Cu, where data with n ≥ 42 are fit with a line. This is because the calculated recombination
enthalpies for Cu2+ with n = 32 and 36 are likely overestimated as a result of non-statistical
dissociation where some water molecule loss occurs prior to or during internal conversion (vide
supra). The linear best fits (solid lines) to the data for all metal ions have R 2 values of 0.98 or
greater. This good linear dependence indicates that ion solvation accounts for the majority of the
decreasing recombination enthalpy with increasing clusters size. The vertical intercepts
correspond to the absolute reduction enthalpies at infinite cluster size, i.e., infinitely dilute
aqueous solution. These values are -ΔHAbs= 1.75 ± 0.09 eV, 2.51 ± 0.07 eV, 3.55 ± 0.10 eV, 1.00
± 0.08 eV for Co2+, Ni2+, Cu2+, and Zn2+, respectively (Table 6.1). A slope of 11.2 eV is
calculated using a modified Born solvation model. 45,47 The measured slopes for the Ni2+ (11.3 ±
0.3 eV) and Co2+ (11.7 ± 0.4 eV) data are similar to the calculated Born solvation model value
but values for Cu2+ (12.4 ± 0.4 eV) and Zn2+ (12.7 ± 0.3 eV) are slightly higher. The continuum
Born solvation model does not explicitly account for quantum-mechanical interactions between
the metal ion and the water molecules, such as ligand-field splitting effects.21 These effects
119
would likely be metal ion specific and may account for some of the variation between the four
divalent transition metal ions investigated here. The higher slope for Zn 2+ may also be a result of
a transition from metal ion reduction at smaller cluster size to formation of an ion-electron pair at
larger cluster size. The same number of water molecules is lost from Zn 2+ as for Ca2+ for n > 50.
EC of Ca2+(H2O)n results in ion-electron pair formation at large n.50 There are also uncertainties
associated with the parameters used to calculate the solvation enthalpies obtained from the
modified Born solvation model, which are discussed in detail elsewhere.45,47
Figure 6.7. Ion-electron recombination enthalpies as a function of n-1/3 for M2+(H2O)n with M = Cu (squares), Ni
(diamonds), Co (circles), and Zn (triangles). The solid lines are linear regression best-fit lines, and the vertical
intercepts correspond to the absolute solution-phase reduction enthalpies for these ions. Data for n = 32 have
previously been published.48
6.4.7 Absolute Reduction Potentials. Absolute reduction Gibbs free energies (ΔGAbs)
are obtained by combining the measured absolute reduction enthalpies (ΔHAbs) with the absolute
reduction entropy term, TΔSAbs:
ΔGAbs = –ΔHAbs – TΔSAbs
(6.5)
Absolute reduction entropy terms have been measured directly or indirectly from isothermal
temperature coefficients (∂E0/∂T) for some metal ions using electrochemical solution-phase
experiments.47,67 Here, an absolute reduction entropy term (TΔSAbs) of 0.49 eV at 298 K for the
one-electron reduction of Cu2+ is used. This entropy term is obtained from the measured
120
isothermal temperature coefficient (+0.776 mV/K), 67 which is combined with the isothermal
temperature coefficient for the absolute standard hydrogen electrode (+0.871 mV/K) and
converted into an absolute entropy using:
ΔSAbs = nF(∂E0/∂T)
(6.6)
where n is the number of electrons transferred and F is Faraday’s constant.45,47 Absolute
reduction entropies or isothermal temperature coefficients have not been measured for the oneelectron reduction of the other divalent metal ions investigated here. The same entropy term as
for Cu2+ is used to calculate absolute reduction Gibbs free energies for Co 2+, Ni2+, and Zn2+. The
absolute reduction entropy terms primarily depend on the type of coordinated ligands and the
oxidation states of the metal ions, rather than the metal ion identity. 47,67,68 From the solutionphase absolute Gibbs free energies, the absolute reduction potentials in solution are obtained
from the Faraday relation:
ΔGAbs = nFE0Abs
(6.7)
where E0Abs is the absolute reduction potential. The absolute one-electron reduction potentials for
Co2+, Ni2+, Cu2+, and Zn2+ obtained with these gas-phase electrochemistry cluster measurements
are given in Table 6.1. The sources of uncertainty in these measured values are discussed in
detail elsewhere, with the error in the modeled REs contributing most to the uncertainty. 45
Table 1. Absolute one-electron reduction enthalpies at 298 K, slope from -ΔHR vs. n-1/3, absolute, and relative oneelectron reduction potentials for Co 2+, Ni2+, Cu2+, and Zn2+. The precision in the slope and vertical intercept reflects
the standard uncertainty in the linear regression best-fit lines.
M2+
–ΔHAbs (eV)
Slope (eV)
E0Abs (V)
E0Rel (V)
Co2+
1.75 ± 0.09
11.7 ± 0.4
2.24
-1.64
Ni2+
2.51 ± 0.07
11.3 ± 0.3
3.00
-0.88
Cu2+
3.55 ± 0.10
12.4 ± 0.4
4.04
0.1669
Zn2+
1.00 ± 0.08
12.7 ± 0.3
1.49
-2.39
6.4.8 Absolute Standard Hydrogen Electrode (SHE) Potential. The one-electron
reduction potential for Cu2+ relative to the SHE potential is 0.16 V.69 A value for the absolute
SHE potential of 3.88 V is obtained from the absolute one-electron reduction potential measured
for Cu2+ in these experiments and the relative reduction potential from solution measurements.
Previous results for Eu3+ using this same gas-phase electrochemistry method resulted in an
absolute value for the SHE potential of 4.11 V. 45 The absolute SHE values obtained from the
Eu3+ and Cu2+ data differ by 6%. The deviation between these two values could potentially arise
from a systematic error in the calculated REs that depends on the charge state of the ions.
121
Laser photodissociation experiments on hydrated divalent metal ions indicate that the
binding energies calculated from the Thomson liquid drop model are slightly too low in this
cluster size range.70 This indicates that the modeled REs (5.0 eV of known photon energies) are
~7% low for divalent product ions.70 There may be an even larger systematic deviation in the
binding energies from the Thomson liquid drop model for 1+ ions used in the calculation of the
REs for the Cu2+ data compared to the binding energies for 2+ ions used in the calculation of the
REs for Eu3+. This would account for the lower absolute SHE value obtained from the Cu 2+ data
compared with that obtained from the Eu3+ data. Laser calibration of these nanocalorimetry data
for singly charged ions would determine whether there may be a systematic error in the modeled
REs as a result of differences in the charge state of the ions.
6.4.9 Relative Reduction Potentials. Relative one-electron reduction potentials for Co2+,
Ni , and Zn2+ have not previously been measured with solution-phase electrochemistry methods,
presumably because the two-electron reduction is more favorable, i.e., it occurs at a more
positive potential. However, the measured absolute reduction potentials for these divalent
transition metal ions obtained with the gas-phase electrochemistry experiments can be converted
into values measured relative to the SHE by subtracting the measured absolute reduction
potential for the SHE. Using a value of 3.88 V for the absolute SHE potential obtained for Cu2+,
the following relative one-electron reduction potentials of -1.64 V, -0.88 V and -2.39 V are
obtained for Co2+, Ni2+, and Zn2+, respectively (Table 6.1). These relative values are expected to
be accurate within the uncertainty of the experiment (~±0.1 V) because any systematic errors in
calculating the REs (vide supra) will be similar for the divalent transition metal ions and should
largely cancel when subtracting the absolute SHE value obtained from the Cu 2+ data.
Even though the one-electron reduction-potentials for Co2+, Ni2+, and Zn2+ have not been
measured previously, formation of the corresponding singly charged metal ions has been
observed with spectrophotometric techniques combined with pulse radiolysis, which produces a
reactive hydrated electron, e- (aq).53,54 The chemical potential for e- (aq) relative to the SHE
potential has been reported to be within the range of values from -2.6 to -2.9 V,71 and any species
with a more positive relative reduction potential than e - (aq) will be reduced by the hydrated
electron. Thus, pulse radiolysis experiments suggest that Co 2+, Ni2+ and Zn2+ must have more
positive relative one-electron reduction potentials than the chemical potential for e - (aq),
consistent with the measured values presented here. Baxendale and Dixon estimated the relative
reduction potentials for Co2+, Ni2+ and Zn2+ as -3.1 V, -2.7 V, and -2.5 to -2.9 V respectively
from thermodynamics relations and by approximating the solvation Gibbs free energies from
Pauling interpolation of the radii and using the Born formula for the free energy of solvation. 72
The one-electron reduction potentials predicted for Co 2+ (-3.1 V) and Ni2+ (-2.7 V) are
significantly more negative than those obtained in our experiments. Recent computed values of 1.05 V to -1.28 V have been reported for the relative one-electron reduction potential of Ni2+.16
These values are significantly less negative than the value estimated by Baxendale and Dixon
and more consistent with our value of -0.88 V.16 The authors attribute the less negative value to a
ligand-field splitting induced energetic stabilization arising from interaction with the water
molecules in the first solvent shell, which can be significant for first row transition metals that is
not accounted for in the estimate by Baxendale and Dixon. 16
2+
122
6.4.10 Reduction versus Ion-Electron Pair Formation upon EC of Hydrated Zn2+.
For n > 50, Zn2+(H2O)n loses the same number of water molecules upon EC as Ca 2+(H2O)n,
which does not undergo metal ion reduction but forms a solvent separated ion-electron pair.50
However, for n < 50, Zn2+ (H2O)n loses about one more water molecule than Ca 2+(H2O)n,
indicating that Zn2+ is reduced at the smaller cluster sizes. EC by hydrated La 3+ results in an ionelectron pair.57 An increase in the number of water molecules lost with increasing cluster size
occurs around 60 water molecules and is indicative of a transition from a surface bound electron
to a more internally solvated electron.57 A similar increase does not occur for Zn2+ (Figure 6.3),
but the -ΔHAbs value of 1.00 ± 0.08 is consistent with the value measured previously for a
solvated electron.57 The relative one-electron reduction potential (-2.39 V) for Zn2+ obtained
from these experiments is similar to the range of values reported for the chemical potential of the
hydrated electron (-2.6 V to -2.9 V71). These results indicate that reduction and ion-electron pair
formation are energetically similar for Zn2+.
6.5 Conclusions
Absolute one-electron reduction potentials for the divalent transition metal ions Co 2+,
Ni2+, Cu2+, and Zn2+ in aqueous solution are obtained using gas-phase electrochemistry of
hydrated ions formed by nESI and ion-electron recombination energies measured using ion
nanocalorimetry. An absolute SHE potential is obtained from the data for Cu 2+, and this value is
used to establish more accurate reduction potentials relative to the SHE for the other ions. In
contrast to traditional electrochemical methods where reduction occurs at the electrode surface
and is controlled by the electrode potential, reduction of these hydrated ions occurs remotely
from the cathode surface and is insensitive to the electrode potential over the range of potentials
used in these experiments. Instead, reduction of the charged gaseous nanodrops is limited by the
sequential capture of electrons. Because capture of a single electron is most probable under these
conditions, formation of the corresponding singly charged ions is readily achieved provided that
these ions are stable in aqueous solution. Co + and Ni+ are stable in these aqueous nanodrops, and
the one-electron reduction potentials referenced to the SHE potential for these ions are -1.64 V
and -0.88 V, respectively. A relative potential of -2.39 V is obtained for Zn2+, but the data also
indicate that formation of Zn+ and formation of an ion-electron pair, consisting of an electron and
Zn2+ in the same nanodrop, are energetically similar. These gas-phase electrochemistry
measurements have the additional advantages that any effects of junction potentials or counterions in traditional electrochemistry experiments are eliminated.
The relative one-electron reduction potentials obtained with these gas-phase
electrochemistry experiments should be more accurate than the absolute values owing to possible
systematic deviations in the calculated binding energies of water molecules used to obtain the
recombination energy values in these nanocalorimetry experiments. These deviations depend on
ion charge state and cluster size and any effects should largely cancel in the comparison of data
for these divalent metal ions. Any influence of the surface potential of water should cancel as
well. Future experiments using laser light of well defined energy to calibrate these
nanocalorimetry experiments should significantly increase the accuracy with which absolute
reduction potentials of individual ions can be measured.
123
6.6 Acknowledgements
The authors thank Professors Peter B. Armentrout and Michael T. Bowers for insightful
discussions and encouragement, and the National Science Foundation (CHE-1012833) for
financial support.
6.7 Supporting Information
6.7.1 Effect of RE on Modeled ETRV and Experimental Distributions. The effect of
RE on the ETRV distributions and the experimental product ion abundances for M 2+(H2O)36 with
M = Co, Ni, Cu, and Zn is shown in Figure S6.1 (a) – (d), respectively. The agreement between
the experimental product ion distributions and the modeled cumulative E TRV distributions varies
with RE, and best agreement is obtained for Co 2+ and Zn2+, which have the lowest RE values.
Relatively good agreement is also obtained between the modeled E TRV and the experimental
distribution for Ni2+ (Figure S6.1 (b)). The modeled ETRV and experimental Gaussian
distributions for these ions differ by less than 0.1 eV in full width at half maximum (FWHM). In
contrast, there is a significant difference (0.4 eV) in the FWHM between the modeled E TRV and
experimental distributions for Cu2+ (Figure S6.1 (c)), i.e., the modeled ETRV distribution is
significantly broader than the experimental product ion distribution. This indicates that nonstatistical dissociation occurs for Cu2+(H2O)36, consistent with previous results for Cu2+(H2O)32,48
where some water molecule loss likely occurs prior to or on the time-scale of internal
conversion.
124
Figure S6.1. Experimental product ion abundances (black sticks) from EC by M2+(H2O)36 for M = (a) Co, (b) Ni, (c)
Cu, and (d) Zn as a function of the sum of the threshold dissociation energies required to form M +(H2O)36-x (bottom
horizontal axis), where x is the number of lost water molecules (top horizontal axis). Cumulative ETRV distributions
(solid lines) are obtained from convolving PTRV,x(E) = (5/2 kT*)-1exp(-E/(5/2 kT*) for each lost water molecule.
Product ion abundances are also fitted with a Gaussian function (dotted line) for comparison. The calculated
recombination energy from the average number of water molecules lost is indicated by the black dashed line.
125
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